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 A005995 Alkane (or paraffin) numbers l(8,n). (Formerly M2916) 8
 1, 3, 12, 28, 66, 126, 236, 396, 651, 1001, 1512, 2184, 3108, 4284, 5832, 7752, 10197, 13167, 16852, 21252, 26598, 32890, 40404, 49140, 59423, 71253, 85008, 100688, 118728, 139128, 162384, 188496, 218025, 250971, 287964, 329004, 374794, 425334, 481404, 543004 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From M. F. Hasler, May 01 2009: (Start) Also, number of 5-element subsets of {1,...,n+5} whose elements sum to an odd integer, i.e. column 5 of A159916. A linear recurrent sequence with constant coefficients and characteristic polynomial x^9 - 3*x^8 + 8*x^6 - 6*x^5 - 6*x^4 + 8*x^3 - 3*x + 1. (End) Equals (1/2)*((1, 6, 21, 56, 126, 252, ...) + (1, 0, 3, 0, 6, 0, 10, ...)), see A000389 and A000217. Equals row sums of triangle A160770. F(1,5,n) is the number of bracelets with 1 blue, 5 identical red and n identical black beads. If F(1,5,1) = 3 and F(1,5,2) = 12 taken as a base, F(1,5,n) = n(n+1)(n+2)(n+3)/24 + F(1,3,n) + F(1,5,n-2). [F(1,3,n) is the number of bracelets with 1 blue, 3 identical red and n identical black beads. If F(1,3,1) = 2 and F(1,3,2) = 6 taken as a base F(1,3,n) = n(n+1)/2 + [|n/2|] + 1 + F(1,3,n-2)], where [|x|]: if a is an integer and a<=x (Matrix([[1, 0\$6, -3, -9]]). Matrix(9, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -8, 6, 6, -8, 0, 3, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Jul 31 2008 MATHEMATICA a[n_?OddQ] := 1/240*(n+1)*(n+2)*(n+3)*(n+4)*(n+5); a[n_?EvenQ] := 1/240*(n+2)*(n+4)*(n+6)*(n^2+3*n+5); Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Mar 17 2014, after M. F. Hasler *) LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1}, {1, 3, 12, 28, 66, 126, 236, 396, 651}, 40] (* Ray Chandler, Sep 23 2015 *) PROG (PARI) a(k)= if(k%2, (k+1)*(k+3)*(k+5), (k+6)*(k^2+3*k+5))*(k+2)*(k+4)/240 \\ M. F. Hasler, May 01 2009 CROSSREFS Cf. A160770, A053132 (bisection), A271870 (bisection), A018210 (partial sums). Sequence in context: A294418 A308669 A115549 * A351643 A034503 A026557 Adjacent sequences: A005992 A005993 A005994 * A005996 A005997 A005998 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu) STATUS approved

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Last modified December 10 19:24 EST 2023. Contains 367717 sequences. (Running on oeis4.)